Cook soppressata and garlic in 2 tablespoons oil in a 10-inch heavy skillet over moderate heat, stirring occasionally, 3 minutes. Add bread crumbs and oregano and cook, stirring, until soppressata begins to brown, 1 to 2 minutes. Transfer stuffing to a bowl and cool to room temperature, about 10 minutes. Reserve 2 tablespoons cheese, then add remaining cheese, mayonnaise, and salt and pepper to taste to stuffing and stir to combine.,While stuffing cools, prepare grill for cooking over low-heat charcoal (moderate heat for gas). (If using a charcoal grill, open vents on bottom of grill, then light charcoal. When Charcoal turns grayish white (about 15 minutes from lighting), hold your hand 5 inches above grill rack to determine heat for charcoal as follows:,When you can hold your hand there for 1 to 2 seconds;,______,5 to 6 seconds. If using a gas grill, preheat burners on high, covered, 10 minutes, then, if necessary, reduce to heat specified in recipe.),While coals are lighting, brush mushrooms on both sides with remaining 1/4 cup oil, then sprinkle with salt and pepper on stemmed sides only. Grill mushrooms, stemmed sides down, on lightly oiled grill rack, covered only if using a gas grill, 5 minutes. Turn mushrooms over and grill 2 minutes more, then transfer with tongs to a tray, keeping mushrooms stemmed sides up. Divide stuffing among mushrooms and sprinkle with reserved cheese. Carefully return mushrooms (stuffing sides up) with a large metal spatula to grill rack and grill until mushrooms are tender, 4 to 7 minutes.